Resolver el sistema dado de ecuaciones diferenciales. dy dx + 3x + dt 1 dt dy -y = e' dt dx dt x(0) = 0, y(0) - 0 %3D USAR UNICAMENTE UNO DE LOS SIGUIENTES METODOS: 1. Transformada de Laplace 2. Método de operadores 3. Método de eliminación

In this project we consider the special linear homogeneous differential equations called Cauchy-Euler equations of the form d-ly aot + a₁th-1 +an-it. +any=0, t>0 dt where ao,..., an are real constants. (Note that in each term the power of the monomial of t and the order of the derivative of y are identical.) A Cauchy-Euler equation can be transformed to a constant coefficient linear equation. So we can find the solutions of such an equation using the methods we know. The procedure is described as follows. (i) Set t = t(x) = e with a being a new independent variable on the interval (-∞0,00). Let u(x) = y(t(x)) = y(e*). By applying the chain rule u(x) = (y(t)) and the fact=e=t, we have the relations dy du 2²y_du dt dx' 1. 32 y dt² dt² + a₂t"-2, (ii) Substituting these relations into the Cauchy-Euler equation we have a constant coeffi- cient linear differential equation for u(r). (Recall u(x) = y(t).) (iii) We are able to find a fundamental set of solutions for the resulting constant…

Use the Laplace transform to solve the given system of differential equations. dx/dt = -x+y dy/dt = 2x x(0)=0, y(0)=7